1. Field of the Invention
The invention relates to the improvement of reticulated structures including truss and lattice types of structures, such as geodesic and related structures. But, it also establishes a new type of structural system, which can be built into shell-type three-dimensional curved lattices and other reticulated, truss-type, structures using prismatic or non-prismatic members, capable of very high load-carrying capacities compared to those of standard three-dimensional trusses.
2. Prior Art
Latticed structures, known as two- and three-dimensional trusses have been built in the past. Usually they were constructed of straight members of various materials, usually of steel, which were riveted or welded together at the joints or nodes by means of gusset plates or by other mechanical connections. Loads were applied at the joints of the upper or lower boom as the structure was supported on some foundation. However, the members in these trusses were kept stocky, not slender or light, in order to avoid what was observed to be the onset of flexural or torsional buckling in the members under purely axial loads. This buckling could not be allowed to occur as such buckling, invariably crippled the structure in a dynamic collapse
As late as twenty years ago this phenomenon in trusses and space lattices of various kinds and, indeed, in light and slender structures in general, was not fully understood and the design codes of the day required that members in a truss were examined, individually for buckling, i.e., as if they buckled alone and not as constitutive elements of a system in which buckling of many members occurred simultaneously under the aspect of finite deformations.
It was thought that only the axial stress in each compressive member mattered and that, as long as that stress or the corresponding axial force remained below a permissible or critical value, the design would be safe. However, there always remained the doubt as to what that value was in a particular case, as discrepancies with the theory were constantly being observed. For pin-ended members in a truss, preselected criteria have been employed, tests have shown that the members collapsed well below the expected values, especially because in actual construction they were restrained at the ends and ideal pins, or hinges, could not be manufactured. In fact, the discrepancies between the failure loads were so large, that for certain slenderness ratios, i.e. the ratio of member-length to the radius of gyration of its cross-section; and, especially, if the members were designed to buckle in the inelastic or plastic range of deformations, it was impossible to predict when the member or column would fail. In this connection various formulas for the buckling or failure of individual hinged members or columns were devised allowing for the initial very small curvatures or eccentricities of loading involved. It was surmised that somehow these were responsible for the premature failures observed as buckling. Only later research revealed that the problem of buckling could be actually explained under the aspect of finite post-critical flexural deformations, causing the displacements of the member-ends or the nodes of a truss-type or other structure to occur and that in slender members the axial elongations or contractions resulting from axial strains were smaller or comparable to the second order displacements of the joints resulting from the contractions of these members due to purely flexural distortions, something which prior analyses never considered. It was then also discovered that minute changes in geometry and the imperfections in the structural elements had a very marked and pronounced effect in the entire phenomenon. This phenomenon has been more fully explained as extensive research into the post-buckling behavior of slender frameworks revealed the actual physical processes and as new physical laws governing the buckling of structural systems in general were discovered and these were formulated into a consistent and now established theory and analysis. Thus, under laboratory conditions, when the loads could be adjusted, it was found that under equilibrium conditions beyond a certain stage of buckling, or in the so-called range of post-buckling, elastic deformations of these structures could be increased as the external loads were being decreased. This paradox has been proven experimentally and explained theoretically for all known categories of elastic structural systems. These deformations remained elastic and the deformation can be cancelled or annulled by increasing the loads in a reverse process of diminishing deformation until the structure practically straightens to its prebuckled shape. See, for example, S. J. Britvec "The Stability of Elastic Systems", Pergamon Press, New York, 1973(450 pages), which is incorporated herein by reference.
Even today, structures of this type are commonly analyzed using the so-called linear theories. These theories assume that the analysis can be performed on the undeformed geometry of the structure in its initial position, or configuration, and that the elastic deformations of the structure do not need to be considered in this process to correct its initial geometry. In addition, it was assumed that these elastic deformations were always linearly related to the changing external loads, which is not true. Moreover, it was also erroneously assumed that, once the buckling deformations had set in and the buckling of the members developed by a finite amount, the external loads remained neutral in this process while the structure persisted in a state of equilibrium. This was also assumed to happen regardless of the type of connections or connectors, used to join the members at the nodes. It was also unknown that other structural parameters, such as the flexural contractions of the members or the parameters characterizing intrinsic geometrical imperfections or the nodal restraints in the structure, have a profound and indeed crucial effect on the form of buckling and on the intensity of the ensuing dynamic collapse and the relationship between these phenomena.
Thus the conventional design methods applied to such structures, especially when they exceeded a certain size, were always confronted with this unpredictable difficulty known as buckling, and its catastrophic consequences in large slender truss-type structures. Indeed buckling had to be avoided at all costs, as the designers wished to avoid or prevent a sudden dynamic collapse of the structure. The correct physical laws which governed the phenomenon of buckling had not yet been revealed, and the proper analyses which could be used to prevent this dynamic collapse by new design methods had not yet been discovered. As a result of the foregoing design problems, such structures were either avoided altogether, or they were overdesigned, that is, made stronger than necessary by keeping the members very stocky, thus resulting in heavy and uneconomical structures. In situations where the structural weight was a consideration, for reasons of cost and/or other modern applications such as structures for outer space, the conventional design methods became inadequate altogether.
Moreover, this buckling phenomenon was previously observed to be accompanied by a gradual development of deformations in the structure well below what was termed the critical range of loading when the onset of collapse followed. This was particularly prevalent in trusses and similar structures, such as portal frames or braced frames, which were built from slender and light members. It was just revealed (see the prior reference) that the resulting buckling deformations were nonlinearly related to the external loads. Therefore, these load-deformation curves or paths became known as the nonlinear equilibrium paths, and the structures which exhibited them as nonlinear structures.
The conventional analyses used before, therefore, did not apply to elastic structures beyond a certain degree of slenderness, or slenderness ratio, in the members in post-buckling. What was missing was the correct understanding of what the limits of the conventional linear analysis were and to which structures it could be applied. Nobody knew at that time how rigidly, or pin-jointed elastic trusses could be analyzed in post-buckling, nor whether equilibrium or an onset of motion in such structures was possible and what forms these can take. Subsequently, research has revealed that a physical process was present for which the governing laws were different than those considered to be operative before. The new theory turned out to be much more complex and it required a great deal of advanced knowledge and insight on the part of a designer or engineer to understand how such structures must be analyzed and what parameters and techniques were relevant for their safe design. One definite conclusion emerged. It was recognized that completely new types of structural systems evolved, once three-dimensional trusses were built beyond a certain size and slenderness of members, and that a totally different response was characteristic for such structures. Some of the characteristics of these relatively light new structures, requiring new and different analyses showed: (1) a gradual development of nonlinear deformations, in states of statical equilibrium, as the loading of the structure progressed; (2) a flexural character of deformation which in some cases was coupled with torsional deformations in the members; (3) a dominant role of the axial forces in the members in which the deformations occurred; (4) a rapid increase of distortions as the loads approached a critical region of loading, depending on many material and geometrical parameters of the structure and very strongly on the type of connectors exercising different end-restraints used to joint the members and a strong influence exercised by the flexural contractions of the members on the physical state of the structure; (5) a marked influence of certain very small geometrical imperfections, incurred in any manufacture of the members and the structure itself, on the shape of these nonlinear equilibrium paths; (6 ) a rapid change of physical state from statical equilibrium to a highly unstable motion which can result in a catastrophic collapse of the structure; (7) and above all, on a great diversity of equilibrium paths depending on the connectors used at the joints and the category of the structure itself, e.g. whether the connectors are soft, rigid, hinged, or the like, and an accompanying diversity of dynamic collapses some of which could be violent.
All this gave rise to various speculations as to how such structures, i.e., slender truss structure, could be made safer. It has become evident that some sort of new optimization techniques had to be devised in order to achieve an optimum design under the circumstances.
Further, modern research has revealed that after the onset of buckling in a perfect structure or in an imperfect structure where this onset is gradual mainly due to the initial imperfections, especially in truss-type structures with quasi-hinged or quasi-pinned connectors, many different modes, i.e. shapes of elastic deformation involving different sequences of buckling members distorting by different amounts, can take place.
It was further found that the buckled structure could be maintained in equilibrium even beyond that state of deformation where the sudden motion developed, provided that the external loads were suitably decreased, and that this decrease was different in different modes and in different types of structures. Thus, different categories of structural systems were discovered in which these modes occurred in very different nonlinear fashions, following the load-displacement equilibrium paths mainly of branching quasi-linear and parabolic shapes. Such structures with parabolic equilibrium paths were apt to develop much more catastrophic forms of collapse than those with nonparabolic and, initially, linear load-displacement paths. It was conclusively shown that the connectors determined what type of nonlinear paths were generated in structures made of slender elastic members, i.e. members with high slenderness ratios, and that quasi-hinged or quasi-pinned connectors were really the key to an optimization of slender trusses, as such connectors permitted the stiffening of individual members in the lattice.
With the foregoing discoveries it has been postulated that there should be an entirely new approach to the design of reticulated shell-type structures, geodesic domes and other more common types, such as slender lattices and trusses. This new approach was no longer limited by only the strength of the material, a normal criterion for design, but rather by the onset of a sudden motion accompanied by large flexural deformations under more or less constant loads corresponding to a critical state, became the governing factor for a safe design of such structural systems. The criteria for such designs, however, vary from one structural category to another. Thus, for example, lattices with hinged or quasi-pin-jointed members are subject to entirely different optimization and design methods than, for example, the rigidly-jointed lattices or reticulated shells or space-trusses. Some of the pertinent methods of analysis are described in the last reference hereinbefore cited. The most important observation in this regard is that these optimization techniques have revealed that dynamic collapse can be totally avoided or its intensity drastically reduced in a prescribed range of loading in trusses and space lattices jointed by quasi-pinned connectors which permit a selective stiffening of compressive members in a critical state, and that, if this collapse occurred at higher loads, its intensity could be drastically mitigated.
Therefore, the buckling process in any slender structure is something very different from what has been imagined. The most important discovery, notwithstanding its complexity, is that this process is strongly and qualitatively influenced by the type of connectors used to joint the members at the nodes and that appropriately designed connectors could drastically modify the course of its response to the external loads in a critical range. The role of minute geometrical imperfections is recognized to be extremely important but, provided these were kept under some measure of control in a manufacturing process, their effect on the statical response of the structure can be accounted for by analyzing the structure as if it were perfect and then, after determining from a model test what value the so called strong imperfection sensitivity parameter had, interpolating the corresponding imperfect path using the theory published in the cited reference, and others, see S. J. Britvec, "On the Nonlinear Behavior and the Stability of Reticulated Elastic Systems", in "Nonlinear Dynamics of Elastic Bodies", Springer Verlag, 1979, also incorporated herein by reference. Thus a verification of the response of any structure of this type can be accurately obtained. This theory is described and applied in the following reference, S. J. Britvec and D. Nardini "Some Aspects of the Nonlinear Elastic Behavior and Instability of Reticulated Shell-type Systems", Developments in Theoretical and Applied Mechanics, presented at the Proceedings 8th SECTAM, April 1976, and in another paper presented at the International Centre for Mechanical Sciences in Udine in Italy in the Lyapunoff Sessions on Modern Problems in Off-shore Engineering in 1980, S. J. Britvec "High Pressure Shells in Off-shore Engineering: "The Post-buckling Analysis of Reticulated Shells", (and most recently in "Post-Buckling Equilibrium of Hyperstatic Lattices", Journal of Eng. Mech. ASCE, 1985 in co-authorship with M. D. Davister), all incorporated herein by reference.
It has been found that there are definite advantages for the methods of analysis described. Optimization and design by which modern reticulated structures can be developed using the connectors described herein, over those built by conventional methods. Above all, the statical stability governed by buckling is thus made controllable and the load carrying capacity reliably predictable, provided, however, that during design, the correct physical laws are taken into consideration. The connectors of the present invention are designed so that the rotational flexibilities in the connectors are developed under the preselected loading conditions. Thereby only those modes of elastic deformation are developed which do not result in a catastrophic dynamic collapse but in which this collapse is mitigated, or controlled or prevented, so that up to a degree of nonlinear distortion, the structure is made to deform in stable or mildly unstable equilibrium. In the preferred practice of the present invention, the hinged action in the connectors never develops in a practical case as the intensity of loading is, by design, not allowed to attain such high values at which this becomes possible, but it is kept below a certain predetermined value dictated by an acceptable factor of safety. Thus, the potential for the development of these rotational flexibilities in the connectors is sufficient to guarantee the physical reality of the described hinge action and, thereby, ensure the stability of the structure in its elastic response.
Thus, the difficulties which existed before in regard to a dynamic collapse of such slender structures can be overcome in the practice of the present invention, by a more thorough understanding of the governing physical laws and by their consideration in the analysis and in the design of the required structural elements, which, depending on the required state of stress and limit loading applied to the structure, make the structure safe for a wide variety of design applications.
The desired hinge action in these connectors is brought about by means of a plastic hinge which is allowed to be induced in the connector in a specific location, where, under the action of the axial load, yielding of the material in the bending mode is produced locally at a grooved neck. This is possible, because at such a neck a certain stress concentration can be brought about due to the geometrical profile of the neck, as soon as the axial force in the member and therefore in this neck, has attained the prescribed intensity corresponding directly to the load applied to the structure externally at its peripheral nodes. These rotational flexibilities or hinges make the adjoining member act as if pinned about the two plastic end hinges situated in the end connectors, so that under the action of the axial force and the geometrical imperfections, it can bow out or buckle as soon as this force becomes critical, which means induced by the external limit loading. If this state is achieved with all those compressive members which when bowing out or buckling simultaneously constitute the desired non-dangerous or stabilized mode of elastic deformation in equilibrium, then the entire structure will deform in a safe manner while stability is still ensured. Those compressive members, which, if they buckled simultaneously, would cause an undesirable kinematic mechanism or mode to develop, are simply stiffened and so prevented from buckling and, as a consequence, the undesirable mode is eliminated. In this manner, only using a systematic approach described later in more detail, the structure can be optimized for its structural stability and a relatively high load carrying capacity to weight ratio.
End connectors which permit the formation of the so called plastic hinges in the ultimate or the critical state of loading are but one device which permits an optimization of a reticulated structure, shell or lattice of the type described. Two and three dimensional truss type structures composed of prismatic or non-prismatic members may be made, according to the present invention, without these flexibilities or rotational freedoms using stiff or semi-stiff, i.e. elastic connectors. However, their response may be very different from that of the systems of the present invention which are made of optimized and plastically hinged connectors, as described herein. That is, they may not be readily optimizable against dynamic instability by the same methods, as described. However, practically any type of truss-type structural systems lends itself to the analysis described herein to design and make structures which will exhibit the described properties, however such structures are highly dependant on the type of connectors used and the methods of nonlinear analysis and optimization in post-buckling vary accordingly.
Further, insertion of additional or redundant prismatic or nonprismatic members into the statically determinate or isostatic lattice of hinge-connected members, having connectors which bear the potential for the formation of plastic hinges under the critical axial forces in these members, makes it possible to increase substantially the load carrying capacity of the lattice. These hinge connected members, when redundant, can buckle simultaneously with the other isostatic compressive, i.e. statically determinate members of the lattice, only if their flexural shortenings are kinematically compatible with one of the modes of post-buckling deformation of these isostatic members. If this is not the case, one or all the modes, comprised of the isostatic members, are prevented from forming at the prescribed level of loading altogether and the loading must be increased further in order that other compressive members within the lattice become sufficiently stressed, and then they may buckle so as to permit the formation of another kinematic mechanism or buckling mode, but now at higher values of the external loads, corresponding to a higher load carrying capacity of the structure. So, if the teachings of the present invention are exploited systematically or methodically, they can be utilized to increase the load carrying capacity of reticulated shells and space lattices. This applies whether buckling is viewed along an equilibrium path, or in a dynamic process or collapse under constant external loads. This also applies to preventing certain buckling modes from occurring in an existing lattice by making it simply statically and kinematically inadmissible.
The process of inserting additional connector-attached members into a structural lattice system to increase its load carrying capacity, can be applied to different types of space elements, such as tetrahedrons combined with ocathedrons, and the like in the formation of space lattices of various shapes which may be curved or flat. Space lattices and reticulated shell type structures can, in the practice of the present invention, consist of polyhedral space elements in which every member serves as the edge of at least two adjacent space elements. Lattices composed of a combination of cube-tetrahedron combination of space elements, have different mechanical properties from those composed of a combination of tetrahedrons and octahedrons, for example. Such arrays of space elements can be combined in larger blocks or super elements and these blocks can be optimized to possess certain desireable properties, depending on the overall geometry of the structure and on the external loading. The most common type of structural forms are dome-shaped structures used for the coverage of large areas, but other shapes, such as hyperbolic paraboloids have gained prominence in more recent times, especially for the construction of large cooling towers. Practically, lattices of any geometrical shape lend themselves to the construction from space elements that will endow such lattices with high and controllable load carrying capabilities.
One other important application of the present invention is to flat or curved large space structures in aerospace engineering applications, such as radiometers, antennas and the like. Such structures must normally undergo complex interorbital maneuvers while they are subjected to considerable inertial loads. The control of these structures in this motion depends critically on the accurate response of the large lattice, as well as on the desired controllability properties which can only be imparted to the lattice if it is properly analyzed and optimized as explained hereinafter.
Three dimensional lattices have, of course, been built before. The structures of Buckminster Fuller, and the Mero-Company in Germany are notable examples. However, none of these structures were designed with regard to an optimal load carrying capacity to weight ratio in the critical and post-critical range of loading and finite deformation using the present invention. In none of these standard structures would the prevention or the control of the dynamic collapse be possible, because such structures are not fitted with connectors which make this possible. The standard methods of analysis and design, used for these structures, do not apply beyond a certain range of slenderness, because they do not consider adequately the finite post-buckling flexural deformations and the end rotations of individual members which govern the response and the statical behavior, as well as the stability of such structures.